Eigenvalue bounds for Schr\"odinger operators with a homogeneous magnetic field

TL;DR

This paper establishes Lieb-Thirring inequalities for Schrödinger operators with homogeneous magnetic fields, providing bounds on eigenvalues that depend on magnetic field strength and demonstrating sharp constants for harmonic oscillators.

Contribution

It introduces new eigenvalue bounds for magnetic Schrödinger operators, quantifying diamagnetic effects and deriving sharp constants in specific cases.

Findings

Lieb-Thirring inequalities for magnetic Schrödinger operators

Eigenvalue bounds depend on magnetic field strength

Sharp constants obtained for harmonic oscillator case

Abstract

We prove Lieb-Thirring inequalities for Schr\"odinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities.